Is there really a correct way to do a squat?

There is a correct way or form to do any exercise. If you do it correctly, it works the target muscles. If you don't and just end up doing something like bending over, you aren't working those muscles and probably just resorting to using muscles that your body is used to using (meaning you don't work as hard, and don't get the desired results).

It can definitely be a big deal for your back (and possibly your joints if your knees are going that far past your toes). It might not seem like it now, but try doing 10 of them after you're carrying an additional 100lbs.

As far as the trainer not being able to explain how to do a squat, he probably should have put both of you in the mirror and you watch as you did a squat in a supposedly incorrect way and then demonstrate better form so that you could compare.

As other people have said, your knees should never go past your toes unless you really want an injury. It's bad bad bad for your knees.
Oh, and this:

if you want a good idea on a proper squat its just like you about to sit on a public toilet but not letting your butt touch the seat.....

Today I went to the gym and had one of those free personal assessments. The trainer asked me to do a squat. So, I squatted and he said I was doing it wrong. Really? A squat is a squat.! He kept telling me to arch my back, so, I'd arch my back and then he'd say...now you're just bending over, so I'd not arch as much and then he'd say my knees were going past my toes. I was about to punch the guys lights out. Seriously! What do you think? Is there a proper way to do a squat? And What's the big deal anyway?

It's alot easier to do them wrong vs right. And probably 90% of the people I see do them wrong. If you want to work the legs and not your lower back and knees, then your trainer was saying the right things. Having the knees in front of your toes puts undo stress on your knee joints vs the muscles you want to actually work. Bowing your back means you are just doing a "good morning" vs an actual squat. ie... working your lower back vs the leg muscles.

Do them right and benefit. Do them wrong and it's a waste of time and you risk hurting yourself.

Why you want to punch him when he showed you do the correct form?
You will damage your knee joints if you don't do them correctly.

Of course there is. When you are squatting at your limit, do them wrong and you will get seriously injured.

Depends if you value your knees and lower back into your golden years....

Your knees should never go past your toes to avoid hyperextension and stress on them. Push your rear back and lower your thighs parallel to the ground, bending at the hips and keeping your core tight to get the maximum effectiveness out o the move and avoid injury.

That's not entirely true. It is more important to have a balance so that the movement is being driven primarily through the hips. The angle(slant) of the tibia should be the same as for the back. For some people, that means their knees will have to go over their toes a little. Think of the "figure 4" pose of a downhill skier.

Also remember when doing a squat your weight needs to be in your heels and not your toes. If you put it in your toes, you wont be getting the benefits of doing squats. You will definitely feel the burn.

Don't know about a proper squat but I do know that you never let your knees go past your toes!

As with everything else I would imagine that there is a proper way to squat to get the most benefit.

Probably a good idea to hold off giving your trainer a punch for a while! :-)