In case there is ever a competition for two ideas that are seen as most different being argued to be compatible with each other...
"Eat drink and be merry for tomorrow you die." That is the general impression that most of us have of hedonism. It isn't an especially reliable impression, though. The basic idea behind it is that only pleasure is inherently good, and only pain is inherently bad, which does not necessarily lead to drunken revelry. Still, I'm not using it that way here. What I am talking about is the idea that what one ought to do and what will bring oneself the most pleasure are identical. A pure hedonist would say that what makes a thing good is that it is pleasurable, but this softening allows that something else causes it to be good and to be pleasurable, and that they therefore always coincide. Thus, a thing is good if it brings pleasure to oneself, bad if it brings pain to oneself, and a thing is pleasurable to oneself if it is good, and painful to oneself if it is bad. So, Pain if and only if bad, and pleasure if and only if good. Note that, whereas utilitarianism is about maximizing pleasure for all who are affected, hedonism is just about maximizing one's own pleasure. It does not necessarily make any claims about what will do that, though.
Stoics ignore the outside world, sleep on concrete pallets, and live on bread and water. At least, that's the image that the word brings up. Again, the principle does not necessarily imply such extremes. Stoicism is grounded in the idea that one's happiness ought not be dependent upon circumstance. You might get unlucky, but why should you be unhappy about something you have no control over? Stoicism does not so much involve an idea of what is good, but it does imply certain attribute that "The Good" must have. By itself, stoicism says that we ought to regard things only with respect to their actual worth. On the whole, stoicism says that one ought to find some basis for one's happiness that will not change with respect to one's ability to gain happiness from it, and that has the most actual worth. Secular stoicism tends to place a person's happiness in the reason we all have at our disposal, as it is the most valuable aspect of our selves, but Christian stoicism suggests placing it in God. ought to Your house burns down? You lose your job? You're charged with, and found guilty of, a crime you didn't commit? Your child dies before you? These things, your house, your job, your reputation, your freedom, your child, are not yours, they are granted as gifts by God. Be glad he had the grace and mercy to allow you to have them for a short time. One's happiness ought not be in one's house, job, reputation, freedom, or child. One's happiness ought to be found in God, who is constant.
At this point the point of similarity ought to be coming into view. Stoicism says to have one's happiness be independent of what happens, hedonism says that the good and the pleasurable are the same. A Christian stoic says one's happiness ought to be found in God, and a Christian hedonist says that what is good will be pleasurable. Thus, a Christian Stoic Hedonist will say that one ought to find one's happiness in God alone, and that this manner of living will give the greatest pleasure to oneself.