Hi Jack,

The oozing sap is known as gummosis, which can be a symptom of multiple disorders. In this case, it's most likely a side effect of the sunscald. When the sap heats excessively in winter, it expands in volume. This can have an effect similar to a pipe bursting, with the sap oozing out of the trunk. Sunscald most frequently affects young, recently planted trees, because they've yet to develop thick enough bark to prevent this type of leakage. In extreme cases, large permanent cracks can develop on the trunk which create an entry point for pests/diseases. This sort of injury is prevented and/or mitigated by applying tree wraps. These wraps are usually crepe paper (don't use burlap), and act as a layer of bark to insulate the tree's cambium (the tree's 'veins' that conduct water/sugars). These wraps typically go on in late November and come off in early April, but it's not too late to apply one. 

https://planttalk.colostate.edu/topics/weeds-cultural-problems/2111-sunscald-trees/

While squirrels may feed on this sap, they more often gnaw at trees to feed on that cambium. This damage can be a problem independent of the sunscald, and potentially a more severe problem. If the squirrel gnaws through the cambium all the way around a branch, that branch will die. You can protect the tree from squirrels by applying a metal barrier around the trunk, but only if there are no buildings or wires close enough for the squirrels to jump over onto the tree.

Squirrels and sunscald are relatively common problems. This behavior of squirrels become increasingly problematic in late winter, and they do it in part because of a lack of alternate food sources. To your point though, this has been an unusually warm winter. Combined with the lack of snowfall, many trees are currently experiencing drought stress. This stress is likely to predispose these trees to other issues this coming season. If you are not already doing so, I recommend giving your tree some water this winter. See our fact sheet on winter watering for more detailed instructions. 

https://extension.colostate.edu/resource/fall-and-winter-watering-of-plants-and-trees/

Let me know if you have further questions!