Well, first I'd have to ask are you legitimately and honestly wanting an answer or are you just looking for confirmation of belief? The reason I ask that is the question and the way you give it is unquestionably pitched in such a way that it displays your own personal bias and belief in the way you term it:
"is it unamerican for the federal government to take my money and give it to someone else because they think they are entitled to it."
In the most basic and literalistic reading of the words you wrote, then yes I'd view that as American. However, what you typed above is in general not the reasons legally or publicly given for why most of the entitlement programs exist in America. You may argue it's the private motivation behind them, but that is arguing what you believe to be peoples thoughts as if its fact, which is a bit more difficult especially when you're making it in a very broad sense.
Now if you ask the following which is a more honest representation of what the supposed purpose of entitlements are:
"Is it unamerican for the federal government to take my money and give it to someone else in hopes of increasing the overall well being of the country."
Then I think you'd get more to the crux of what entitlement programs purposes are. In that case I think it becomes far mroe questionable whether its "unamerican" to do such. Essentially the view on one side that no matter what "Good" you may think may come from the end, the means of taking it from someone that earned that money for himself is not justified by it. On the other side you have the view that the individual losing the money is still benefiting from it by people part of a society who overall is improved.
So to see if this is "unamerican" or not we need to look at the very principle of it...having something taken from you to help someone else which might or might not have a directly positive affect on your life. This is essentially what the government does often in a general sense of some sort. It is not "unamerican" to have taxes, there was nothing inherently "anti-all taxes" within the founding documents or the founders. They realized that government is an entity and anarchy is not preferable, and as such a government entity needs revenue of some kind. This money generated by taxes goes to variety of things like roads, the military, etc. Many of these things, especially in early America, could be very questioanble about rather or not they directly affect an individuals lives and how much it affects their lives in an indirect way.
I think Entitlements are generally an extension of this kind of thinking. As such, I think in the end, "no", I do not think Entitlement programs on the governments side of things are necessarily "UnAmerican" in principle. You could make an argument that the notion that it "helps society" isn't true, but that's an issue of opinion or disagreement on the outcome and not necessarily about intent. I do think that you could have a case in regards to some/many of them of being potentially unconstitutional, but something that is unconstitutional or someone that promotes something that's unconstitutional doesn't make someone "UnAmerican" (If it was you'd have George Bush being "UnAmerican" amongst almost all of the 2001 congress). However, in general, the notion of the government taking a citizens money to put that money into programs that's going to primarily help other citizens but overall believed to help everyone is not in and of itself an Unamerican theory.