Not true. In Texas, a public place is one "...to which the public or a substantial group of the public has access." Typically, a Wal Mart parking lot will not have a controlled access gate or a parking attendant screening who is allowed onto the lot, and you don't need a parking pass, which makes this a public place, per the Texas definition. Heck, you can get a DWI charge for driving in that same parking lot.
Now, I am no expert, but I don't see a justification for the officer's actions, unless there was something that happened between the officer and the 2 young men prior to what we see in the video. Barring that, if there are no signs that prohibit video-taping, then there is no expectation of privacy for the store itself. The OFFICER can request that he not be video-taped, as can any person. But in and of itself, this is not a criminal act (to my knowledge) and the only remedy I know of for filming someone without their consent is in civil court, and they likely wouldn't award any damages if the person who shot the video isn't making money off of it.
Absent of probable cause, I don't think the officer is justified in asking for an identification, as there was no criminal offense.