I think not. First of all the obvious: $p\iff q$ always implies $p\implies q$, but not the other way around. 
Since beginning students may not have that totally nailed down yet it's better for them to err on the side of $\implies$ rather than mistakenly write $\iff$. 

Now about extraneous solutions, I think that technique wouldn't help much anyway because the point is that if you square the equation and get two solutions $x_1$ and $x_2$ you still need to *check* which of them satisfies the original equation, whether you are aware of the "broken" $\iff$ relation or not. So actually you are more likely to check if you are aware that it is $\implies$ all the way through, since it is a reminder that the equation implies the solutions, but *not* the other way around.

In other words, I think what's more important is to understand that when we solve an equation, we are interested in the solution, and therefore the direction of $\implies$ is what interests us. For that exact same reason, we need to check which of our solutions satisfy the equation if we get more than a single one. (Of course, at the more advanced stage the student would know ahead of time how many solutions to expect, but I'm assuming the question is not concerned with that level of math yet).