Here is the picture of it I took with my phone while I was out working:
Here is a close up:
Location: Texas (winter).
Your image is not the best one, but this is probably the Magnolia green jumper, Lyssomanes viridis.
Here is a better image of it for comparison:
It also matches your location. According to the Wikipedia page linked above,
the species is native to the Southeastern United States plus Texas, and as far north as Maryland.
I'll go with the jumping spider, for two reasons - first, the Lynx Spiders make a normal brown egg sac, and second, there is a strikingly similar jumping spider in Australia, called Mopsus mormon, that also makes a green egg sac (although, to be fair, it's the eggs that look green rather than the silk..) http://malcolmtattersall.com.au/wp/2015/10/mopsus-mormon-mummy/