If I do pushups, pullups, and squats (bodyweight only), will I build ANY muscle at all?

Yes, yes you will. As you said, throwing weights into the mix (even free weights) would help your cause dramatically. But you would gain muscle mass from doing those exercises, so long as you maintained enough protein and calories to do so.

Push ups are a great way to tone your arms and chest as are pullups.

Squats will help with your legs, butt, and core.

If you can afford it, getting an exercise band and you can do to help with some resistance.

Absolutely! I would recommend calisthenics and isometric exercises early in any exercise program to get your body started building muscle. If you jump right in with weights and use improper form, it is very easy to injure yourself and lose any previous gains while you recover..

Yes, you will. You could buy used weights or maybe even pick some up on craigslist for free.

Definitely. Throw some rocks in a backpack for added resistance.

I was thinking some more, you can use cans of fruits and vegetables, milk jugs and even cat litter buckets for weight before graduating to formal weights.

I have used 2 liter pop bottles (full, of course) but you can buy solid weights of various sizes for not very much money.

You could also get a job in the Tree industry, carry logs all day. Really sorts you out.

Yes. Make sure you are getting enough protien, so you muscles can build.

Yes.

Absolutely. Those exercises use many muscle groups together instead of isolating them like stupid gym machines. Plus if you do lots of them quickly you’ll get a solid cardio benefit.

Try this workout:

complete as many rounds in 20 min of
-5 pull ups
10 push ups
15 squats

Or do this one, The Seven Minute Drill:
Seven rounds as quickly as possible of
-10 sit ups
-10 squats
-10 burpees (google burpee if you need to)

If you don’t think either of those provide a solid workout that leaves you sore, let me know.

I’ve done what @cockwain lined up in the 20 minutes and it kicks your butt.

You can absolutely gain muscle this way, providing that you eat right. You need to eat more calories than you burn off, and get plenty of protein in there.

Pullups and pushups are actually a great way to build upper body muscle. Bodyweight squats aren’t so good because your legs are already used to hauling your weight around all day, but they will be beneficial. I would suggest trying Ski Squats: http://www.abcbodybuilding.com/exercise3/skisquat.htm

This will really burn your quads. The page I linked says that you should do 1–5 seconds in each position, but I like to do more like 20 seconds. You can do longer or shorter times depending on what you’re capable of. If you can work up to 30 seconds per position you’ll REALLY be killing your legs, which is great for building muscle.

There are a whole load of exercises you can do with body weight alone. Just do a google search for “bodyweight exercises” or “bodyweight exercises for mass” and you’ll get loads of results. There’s a pretty good list here: http://shahtraining.com/top-9-bodyweight-exercises-for-mass-gain/ but there are loads of variations on exercises using bodyweight to work different muscle groups if you look for them.

Totally agree with what @WhatEvil is saying. Note that his second link recommends handstand pushups, and if you follow that link there is a Crossfit video for doing a handstand pushup at the bottom of that page.