I have my players come up with a bit of their character backstories, but just like their character builds, they must be approved. In general though, the only reason I would not approve a backstory is if it doesn't really fit into the fluff and/or theme of the campaign.
I do not change their backstories without telling them, but I will disapprove of one and make several suggestions on how they can make it fit the fluff and/or the theme of the campaign.
For example, on Golarion, all androids look similar to humans. Though it would be possible to make androids look like half-orcs, noone on Golarion (not even the Technic League) has the level of knowledge to do so (perhaps in the future). If the player wants his android character to be built by his "parents", in the shape of a half-orc, and have them over the years build new and better bodies and transfer the soul from one android to another multiple times... well I'm sorry, that just doesn't fit the fluff. Like not at all.
Also the campaign is a story about a party of adventurers, and no one player character should have such a special snowflake world-changing background that the plot becomes focused around that character. This would take too much spotlight away from the other character arcs.
In the above example, if someone on Golarion really did find their own way to make a new kind of android, and transfer souls from one body to another, the plot should really center around the Technic League (and the androids of Szamrak's Haven) doing whatever they can to find this creator and his creation. And then the half-orc android created by the player should really be an important NPC and not a PC.