No, and rich well-off people aren't happier than poor people. That is all based on essentialist bullshit. Proponents of the working class and poor, as well as those who romanticize austerity tend to talk about it as if it is liberating, or as if the poor are somehow morally superior to the middle class and wealthy.
Those who idealize accumulation tend to talk about material comforts and wealth as ways to access happiness (as an aside, I have rarely met people of this type who actually argue that wealth and material comfort = more happiness, rather that it provides opportunity for more happiness. I think this is a place where often anarchists get it wrong in discussing this sort of stuff). Certainly, to an extent this can be true (as can the opposite - that a lack of things, or attachment to things can help one live more intentionally, to sound like a Buddhist for a moment), but I don't think it is possible to generalize.
In regards to people who talk about the poor being happier, I actually think this serves to marginalize them as a class in the same way that stereotypes of the simple and happy black people free from the cares of the white world (zippety-doo-da!) has done so to black folks.
Also, there is a part of Nihilist Communism that I think goes well with this. I have some criticisms of the book as a whole, but really like certain parts, including this: http://theanarchistlibrary.org/library/monsieur-dupont-nihilist-communism#toc17
(edited for minor punctuation)