The big thing that you you are going to see cooking in regular pans vs. non-stick is the addition of pan sauces to your table. It is almost impossible to get a pan sauce out of the non-sticks because they prevent the formation of fond in the pan. As for stick in a regular pan, it is not much of a concern for most items, but there are a few things that non-stick are invaluable for so I generally keep a cheap non-stick around for those. (Talking cepes and eggs, mostly) The biggest trick with regular pans is learning to stop messing with your food while it cooks. You want to let it form that crust which as a by-product will help prevent sticking. Also, you will find that you start at a higher tempature when cooking with regular pans.
There are two parts to this question, the stated part, and the unstated "are you really frying an egg if there is no oil?"
For the first part, most manufacturers of non-stick pans claim that their product makes oil unnecessary, and generally I've found that to be true. A little oil helps, but "necessary" might be a stretch.
To maximize your non-stickyness of a non-stick pan, you should always preheat the pan before adding the food, at least to 250F or so. Preheating causes the teflon to expand a bit, closing up pores and micro-scratches in the surface.
It is also common for oils from the past to form a polymerized layer on top of the teflon, making the pan more sticky. This is actually worse when you use aerosol spray oils, because the tiny droplets hit the pan and very quickly polymerize to form a slightly sticky layer. This layer is similar to the one you intentionally create when seasoning a cast iron pan, but in this case it is not helpful. All-Clad recommends periodically cleaning teflon pans with a light scrub of baking soda and water to remove that coating of oil. It should bring back the maximum slippery of your teflon.
When cooking protein-rich foods, you can also minimize sticking by letting the food cook a bit before attempting to move it. The loose, floppy denatured proteins in an egg are extremely sticky, but when they coagulate (set) from cooking, they become less sticky. It may seem impossibly sticky at first, but may release on it's own after a minute of cooking.
As to the second, unstated part, oil provides flavor, and is also a heat-transfer vehicle. When the food is on the pan, if you looked at it with a microscope only a small portion of the food is actually in contact with the pan. Those areas will get heated more, and other areas will get heated less. A little oil fills those gaps, causing more even thermal contact, which allows fried food to fry, rather than just get burned in some areas.
It is really a matter of taste whether the flavor of the oil and the way that it changes the texture of the food is appealing to your boyfriend. I wouldn't like it without any oil, but if he does, well, so be it. If he doesn't really like the taste, but is concerned about health, I'd suggest using a little olive oil - it has no cholesterol, is low in saturated fats, and has a good flavor. Different than butter, but still very enjoyable.
Best Answer
All Teflon pans are non-stick, but not all non-stick pans are Teflon ;-)
Teflon, when new, is one of the slippiest substances known to man. Nothing will adhere to it, not even oil. When new, you can have a hard time getting an even coating of oil unless you make it deep enough to fill a base layer. The oil will be more attracted to itself than to the pan.
Once you've been frying in Teflon for a few months, the surface gradually does begin to get coated in carbon, from burnt ingredients &/or oil. I don't know quite what causes this - potentially the surface getting slightly damaged or roughened by abrasion; but an older Teflon pan will more easily coat in an even layer of oil.
If it pools in patches, then the best reason I can think is that some areas are more abraded than others, or the pan base may be going out of true. Most cheap frying pans will eventually bow one way or the other over time - some forming a 'moat' round the edges as the base has bowed inwards, others a 'pond' in the middle, as it bowed outwards.
A 'quick fix' for cheap pans affected in this manner is to put it [cold & clean] on a carpeted floor & gently stand in it. It's a bit rough & ready but it's not worth fixing professionally.
From comments - The Frying Pan Ballet Company…
[credit David Hoffman + 10 mins in Photoshop by yours truly]