You've touched on why it's problematic for educators to only talk about proving in the context of formal proofs. Students need to be accustomed to mathematical reasoning and justification well before they ever see a formal proof.
This is the reason that Common Core State Standards Mathematical Practice 3 exists:
CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the
reasoning of others. Mathematically proficient students understand and
use stated assumptions, definitions, and previously established
results in constructing arguments. They make conjectures and build a
logical progression of statements to explore the truth of their
conjectures. They are able to analyze situations by breaking them into
cases, and can recognize and use counterexamples. They justify their
conclusions, communicate them to others, and respond to the arguments
of others. They reason inductively about data, making plausible
arguments that take into account the context from which the data
arose. Mathematically proficient students are also able to compare the
effectiveness of two plausible arguments, distinguish correct logic or
reasoning from that which is flawed, and—if there is a flaw in an
argument—explain what it is. Elementary students can construct
arguments using concrete referents such as objects, drawings,
diagrams, and actions. Such arguments can make sense and be correct,
even though they are not generalized or made formal until later
grades. Later, students learn to determine domains to which an
argument applies. Students at all grades can listen or read the
arguments of others, decide whether they make sense, and ask useful
questions to clarify or improve the arguments.
You can see important foundations of formal proofs in this mathematical practice. Practices are meant to be developed throughout the mathematics curruculum, across all grade bands (at whatever level is appropriate for the students).
Ways that some of these practices can be addressed are found in many places. For teachers with students before secondary school I suggest looking at Developing Essential Understanding of Mathematical Reasoning for Teaching Mathematics in Grades Pre-K-8. Central ideas of this book are conjecturing and generalizing, along with justifying and refuting, and how they can be a part of the mathematical activity of students in the classroom.
If you're getting the idea that I am not suggesting a quick fix, you've got the right idea. I really think that students should come out of their public school education realizing that mathematics is something other than calculation -- that the mathematical practices themselves are ways of thinking that help you understand and act in the world more effectively.
So, my ideal solution is to be adamant that the mathematical practices are a focus of school mathematics. That way students will have a chance to know why conjecturing, generalizing, justifying, and refuting are important.
Okay, but how do you convince people in "everyday life?"
There is another book in the Essential Understandings series that deals with proof and proving for secondary school age students: Developing Essential Understanding of Proof and Proving for Teaching Mathematics in Grades 9-12. This is not a book on "how to do a proof" but, rather, how and why proofs and proving are a part of mathematics and life. I wouldn't say it addresses your question directly (although, perhaps if you consider everyday people as similar to secondary school students, there is some utility here). Even so, I like these ideas from the book:
Big Idea 4: Not all arguments are proofs. A proof is not an argument based on authority, perception, popular consensus, intuition,
probability, or examples.
A good deal of the rest of the book talks about what proof is, but this idea about what it isn't may be especially helpful when explaining to people that proof has a unique place in mathematical reasoning that is not fulfilled by these other sorts of arguments.
Why do we need this aspect of mathematics? The role of proof is discussed in Big Idea 5's subcategories.
- to verify the truth or falsehood of a statement
- to provide insight as to why a statement is true
- as an entry point into the development of a new idea
- as a structure for communicating mathematical knowledge
- as an impetus for the use of precise mathematical language
As an example of how that last one can be demonstrated, I worked on a math ed technology research project that used software to collect a whole classroom's piecewise-defined position vs. time functions and graph them on one Cartesian space. Students created narratives to go along with their functions (the context was an "exciting race" in which all the participants run for the exact same amount of time, have the same start and end position, but all have different ways of completing the race). One student is chosen to read his narrative, and then the whole class tries to pick out his graph from all the other ones.
Generally, it is difficult to chose one such graph out of the 30 or so student functions. And, usually, it is because their narratives lack mathematically precise language. So this becomes an opportunity to show how precise mathematical descriptions communicate much more than a loose narrative, but also how those mathematical statements can be used to justify the conjecture that "Sam's race is the red graph." "Why?" "Because he ran 6 meters/second for 10 seconds, then stopped for three seconds, and that's what the red graph represents." "How do you know the red graph represents that?" Etc. It is much harder to make a convincing argument without the mathematical language and the precision it brings.
So, my advice is to come up with examples that fit into the categories listed above in the role of proof. Cater the examples to what your audience cares about. But if it's students, there are some decent resources. Look specifically to resources that focus on mathematical practices (especially CCSS MP3).