I'd like a citation on state law that says an employer can search your person or your belongings without your consent, even on company property. Now, company policy may say one thing, but if they're not LEO and they don't have a warrant, they're not going in my backpack. If they attempt, there will be a physical altercation to prevent their illegal search of my personal property.
Now... they can fire you at will if you refuse, of course... but I have never seen anything saying it is legal for them to seize your personal property for a search. Which would include taking my backpack away from my personal control. I could be wrong, but like I said, I'd like to see a state law that says so.
<edited to add> Lockers I see the same as a company vehicle - their locker, they can search. But they're not going inside a backpack or something in my personal possession, any more than they're going inside my wallet or my undershorts. I'm jus' sayin'.
It all depends. For instance: My company provides a tool bag for me to use on the job, along with providing me all the tools. However, after 12 years of carrying said tool bag my back started to give me problems (carry 40+ lbs in your right hand for that time period and you'll understand). So, I bought myself a technicians backpack. Now, the backpack is my property, but I'm using it to house my company's tools. Therefore they have the right to search my backpack to inventory the tools.
The same could be said for a toolbox if you provide your own and it houses company owned tools. So its not a simple "yes they do" or "no they don't".