I have moved so MANY times in the past several years, here's my suggestions.
NEVER use boxes that are used for food items, such as ones from grocery stores. Often these boxes are stored in refrigerators and have been subjected to moisture which softens the fibers. They almost always break.
Check out Craigslist.org - people move all the time and a lot of people advertise their used moving boxes, some free, others for a fee. Just check back everyday. Also consider placing an ad for boxes on craigslist too.
Ask neighbors. Have you had someone move into your neighborhood recently? If so, just ask, even if you don't know them. You may be surprised. They might have tons of boxes sitting around, but just haven't found time to get rid of them.
Drive around your neighborhood on recycling day. In my city (Plano), people can put their flattened boxes out for recycling pick up. Pick them up provided it's not raining or wet/damp.
If you don't find enough boxes or ones that meet your requirements, you can buy boxes reasonably at Lowes or Home Depot.
I hope this helps!
K.
SAHM to 3 ages 7, 5 & 3