According to Cohen, the Packers do (or did) want him for the PS:
"Though White was waived Monday, cutting short his third attempt to stick on the 53-man roster, the Packers would like to sign him to the practice squad, according to a source. To do so, White must first clear waivers and then decide if he is willing to remain in Green Bay."
http://www.jsonline.com/blogs/sports/325435201.html
That would stand to reason. White was chosen for the 53 over a healthy Pinkard in the first place, whereas the latter player currently resides on the PS. Abbrederis is in a different situation...the original perceived potential has not yet been explored due to injury.
I think it quite plausible that White would want to test the waiver wire for a 53 roster position, bide his time for a week or two as the injury count around the league accumulates, or pursue other PS options.