first of all i would suggest looking in a retail store to find the model you want first. you ALWAYS want to view the screen in person if at all possible first.
second if you are intending to game with it there is no reason for 120 or 240hz. tvs only take a 60hz input anyways and the extra frames are faked which causes lag in games (read up on this if you dont believe me). you WILL be using 60hz game mode anyways so why buy something you dont need. with that said, its getting harder and harder to find 60hz models anymore.
$700 for a 40" should be sufficient for base models of respected brands like sony, samsung and lg. you could go second tier with a toshiba, rca, vizio or other brand but parts availability could be an issue if it ever breaks on you. this is why i tend not to recommend anything but the big 3. i would completely avoid things like olevia and other no-name cheap discount brands. junk through and through.
must it be from newegg? you may be quite limited on selection. it may be better to use amazon or price match amazon in a retail store. unless of course you have newegg cash (gift cards, etcetera)
personally i'd suggest a good 60hz sony with the bravia engine without any smart-tv and any other garbage functions. i've used one for 5 years or more and it works great. as a second choice i would say perhaps a samsung in 60hz. as a third choice i would say either a sony or samsung 120hz but with a working game mode since you will be running it in 60hz mode all the time for gaming.