According to this post on the Cryptomundo website, Texas wildlife officials have opined that it would not be illegal to hunt and kill a Bigfoot and/or Sasquatch, should one be located (and shot) in that state. In response to an inquiry, the Parks and Wildlife Department wrote:
If the [Parks and Wildlife] Commission does not specifically list an indigenous, nongame species, then the species is considered non-protected nongame wildlife, e.g., coyote, bobcat, mountain lion, cotton-tailed rabbit, etc. A non-protected nongame animal may be hunted on private property with landowner consent by any means, at any time and there is no bag limit or possession limit.
An exotic animal is an animal that is non-indigenous to Texas. Unless the exotic is an endangered species then exotics may be hunted on private property with landowner consent. A hunting license is required.
Why was someone inquiring about this? It appears that there is currently quite a controversy among members of various Bigfoot-research organizations as to whether it would be okay to shoot a Bigfoot — some argue this should be done in order to collect a "specimen" to confirm it as a new species, while others oppose it. This seems to have come to a head after an incident in 2011 in which some Texans "attempted to collect [a Bigfoot] for scientific analysis," although it sounds like they might actually have been shooting at somebody's nephew (who, luckily, they failed to collect). Following that incident, the Texas Bigfoot Research Conservancy (one of many such organizations, actually) said it would not rule out trying to bag a Bigfoot in the future, which prompted two members of its board of advisors to resign. The inquiry above was apparently made by one of their allies, who also favors a "no-kill" policy.
Some commenters have asked whether Bigfoot would be protected under federal law, and the answer to that question is apparently "not right away." This is based on a 1977 press release by the Department of the Interior addressing a similar concern over what would happen if a Bigfoot was actually found, and whether he/she/it would be protected under the Endangered Species Act. Certain procedures would have to be followed, the press release stated, though the Secretary could put Bigfoot on the endangered list for 120 days on an emergency basis. Long-term protection would involve "the same regulatory mechanisms already used in protecting whooping cranes and tigers," but that obviously won't do the first specimen any good.
Be advised that you also need to check local law. The best-known Sasquatch-protection ordinance seems to be the Sasquatch Protection Ordinance of Skamania County, Washington. Passed in 1969, this made it a felony to kill a Sasquatch, punishable by a $10,000 fine and up to five years in jail. Reportedly, that was reduced to a misdemeanor in 1984, at which time a "Sasquatch refuge" was also created in the county. As this table shows, neither ordinance was codified, but it appears that they were actually adopted and seem to still be in effect. The county did adopt a new code in 1979, but if that replaced all prior ordinances, then there would have been no need to amend the 1969 law.
The bottom line is that you should probably check with local authorities before going out to try to murder a Bigfoot, even in Texas. Better yet, you should probably just leave Bigfoot alone, especially since it probably doesn't exist and so you're most likely shooting at somebody's nephew.