While I can kind of "see" what some people mean by using terms denoting a race is singling them out and promoting racism, there is more to it than that. The way the term is used may be singling out a race, but depending on how it is used it can either be a matter of awareness (positive, hopefully) or discrimination / superiority (negative). In the case of Black Lives Matter, they are saying Black lives matter too... they are not trying to imply only black lives matter or no one else's lives matter, they are saying "look at what is happening to us, we matter!". In this case it is a matter of awareness, not promoting a race as better. When you look at it from this perspective and someone says "all lives matter or its not just black people" then you are essentially negating the whole purpose of what they are saying.
However, I do believe that there is a lack of understanding how some people think. Not everyone has the same background, and nothing at all will make everyone agree or be happy. Which is ALSO discriminatory. If someone goes off stating how using something like X is also racist, severe punishment and firing them will just amplify their misunderstanding and promote more hate. Maybe looking beyond the words to see what they are actually implying instead of jumping at them would have a more positive effect. I am not saying there are not racist issues out there.. there are many, many many but there are some people that either do not understand the context of what is being said, or do not have appropriate means to express their thoughts which should have the right to express their thoughts as well... just without the poor choice of words.