There's nothing specifically wrong with leaning right or having a bootstrap mentality, though. There is a degree of truth in overcoming obstacles yourself rather than expecting help and it's not like everything conservatives stand for is automatically wrong, either. In fact, most of the time, it's the people who represent conservative values that are the issue rather than the values themselves.
As for the snowflake thing, it's kind of a widespread issue independent of political ideology. Even in the field of comedy, lots of jokes no longer fly because people are now deciding they're offended by them. It's very difficult to draw a line between things that are actually offensive and things people are choosing to be offended by and it's very easy to get dogpiled for enjoying things the internet has decided in the last few years is no longer acceptable. Take Mel Brooks' movies - it's frowned upon to like even Young Frankenstein or Spaceballs these days because they're considered offensive and Mel Brooks himself admitted he couldn't release those movies today because they'd be considered too offensive.
And it's not just a "millenial" thing, either. Remember last year and the whole punching Nazis thing and how conservatives got super offended by it and demanded civility?