A Walmart Schwinn might be fine for tooling around town but it is definitely a low end bike. Using such a bike might be an interesting strategy for a cross country trip. When it breaks down, you just throw it away and buy another from the nearest Walmart!
Seriously, if I were planning a cross country trip, I'd invest in a good quality bike. The extra cost for the bike is minimal compared to the total cost of the trip. May as well get a bike that will be comfortable during those many, many miles in the saddle and be less likely to break down 100 miles from the nearest bike shop.