Can animals like squirrels leave "gifts" or tribute for humans?
I believe a lot of these behaviours can fall under the umbrella term of reciprocity, or  reciprocal altruism.


  In evolutionary biology, reciprocal altruism is a behaviour whereby an
  organism acts in a manner that temporarily reduces its fitness while
  increasing another organism's fitness, with the expectation that the
  other organism will act in a similar manner at a later time (...) The concept is close to the strategy of "tit for tat" used in game theory.


For example, red-winged blackbird males help defend neighbor's nests. There has been some discussion around the reasons for this behaviour, but it has been observed that males reduced the amount of defense given to neighbors when neighbor males also reduced defense for their nests. Since kin selection doesn't seem to be an important factor here, this would support a tit-for-tat strategy (animals help those who previously helped them). 

Some primates do this frequently too:


  A monkey will present its back to another monkey, who will pick out
  parasites; after a time the roles will be reversed. Such reciprocity
  will pay off, in evolutionary terms, as long as the costs of helping
  are less than the benefits of being helped and as long as animals will
  not gain in the long run by "cheating" – that is to say, by receiving
  favours without returning them.


The example with the crows, mentioned in the comment, can be interpreted under this light too. Those crows seem to give presents expecting something in return (albeit delayed!). 

What's interesting though is that altruism, at first sight, would seem to go against reproductive success. Group Selection, however, is something Darwin was already talking about in Origin of the Species.

So, back to squirrels. I didn't find particular research about them in their relationship with humans (will keep looking), but analysing it under reciprocal altruism it could be justified that they give nuts expecting a potential future benefit for them or their group. There's much more to this complex dynamic, but reciprocity is definitely a strong candidate for explaining your kind gift-presenting squirrels. 

Some quick references and materials:


The evolution of reciprocal sharing
The Gift, by Marcel Mauss (complete online book) --> From an anthropology point of view, Mauss was the first one to talk about recicprocity as economics. Absolutely recommended read.
Social semantics: altruism, cooperation, mutualism, strong
reciprocity and group selection