We need to be very careful about language with this question. There are multiple ways that we could think about this:

{Can,Should,May} {you in particular, one} {prepare as an edible dish, heat up so that its physical state changes, hard boil, scramble} {a chicken’s egg, an ostrich egg, a Cadbury’s cream egg} in {Johnson’s baby oil, the oil made from the fatty secretions of an infant}?

I make it 144 possible readings of this question at least, and I’m sure that there are many more.

I should be quite clear at the beginning that for just about all of these it is a bad idea, and the answer is you definitely shouldn’t.

We will take only the premise that it is something akin to Johnson’s baby oil that the question is referring to. Some versions of this form of baby oil do contain a small amount of oil which is akin to vegetable oil, however alongside this ingredient are others which are definitely not edible – at least in quantities of more than a few milligrams, and even that I would not recommend. In fact Johnson’s baby oil in particular does not contain vegetable oil at all. Other brands do.

So, cooking anything which might be tainted with the oil is definitely a bad idea.

One could conceivably boil an egg in it. However, shells tend to have some porous nature, and so there is a chance that one could still taint the egg even if you were heating an egg in its shell in the baby oil, and this would probably not be a good idea.

On the purely practical side, it is likely that you could get baby oil above the temperature needed to change the protein structure of an egg, but again, don’t do it!