You ask a fair question and I suspect that some of the answer, in the case of the squirrels, is that the red squirrels are thought to be more attractive than the grey squirrels, so many people would prefer that the grey squirrels do not take over. But there are also reasons for environmental concern (i.e. not just aesthetic preference) in that there is reasonable fear of a loss of genetic diversity and also fear of upsetting the ecological balance. Invasive species cause the extinction of other species without replacing them with other new species, hence loss of genetic diversity (and vulnerability to future environmental challenges). Invasive species also often upset the ecological balance e.g. rabbits in Australia and then cause the extinction of several species, not only the species that they directly compete with. Grey squirrels are thought to damage the woodlands in England, and thereby the environment of some bird species.
The claim that something "symbolically supports tyranny..." is not a claim about the act in itself, but a claim about the meaning of the act. Your vegan friend may see troublesome meanings in the act of eating artifical bacon flavored chips or wearing fake fur. But not everyone does, and there is much ambiguity and complexity about what things mean. To take another example, I have friends who will not marry because they think that "marrying symbolically supports an institution that oppresses women." I don't doubt that marriage has historically oppressed women, but I think marriage has multiple meanings (commitment, family, for example) and any decision whether to marry or not needs to take all these meanings into account. Back to the fake fur: Personally I prefer fake fur that looks fake, so that I don't make unnecessary enemies or set a bad example. Your friend is so passionate about veganism that he focuses on one set of meanings.
Humans have both rights and responsibilites, but other beings may have either, both, or neither. Think of infant humans: they have rights but no responsibilities. As children mature, they get some responsibilites. Mentally disabled adults have rights, but sometimes not the full range of "normal" adult responsibilities. Responsibility depends on the capacity to distinguish right from wrong. Having rights depends on (most people think) being sentient.