The Apple warrantee extension program only refers to video and power problems, not freezing. The most sure-fire way to check if you have bad caps is to open the access door at the back of your eMac and look at the logic board next to the RAM slots. There are two little objects that look like 1/4 of a AA battery. (Googling "capacitor plague" should get you some pictures of capacitors if you're not sure what to look for.) If the top of either or both of them is cracked, swollen or leaking a discharge you know for sure that you have bad capacitors and Apple should have no problem fixing things for you. If you don't see any of those things, it might still mean that you have bad caps but they haven't shown it yet.
If you do plan to open your eMac, it's a good idea to put a towel on the desk so that you can avoid scratching the screen. It makes it more secure, too, and no doubt more comfy for your eMac.
eMac 1.25 GHz Mac OS X (10.4.7)
