Is there a legitimate reason I should be required to use my company's computer? (BYOD prohibited)

So this is an interesting question with a few points into why you not only should WANT to do this, but should do this for your own safety and security. It helps first if you understand that companies point of view before we talk about how it can benefit you.

Why would a company want to do this?

Many reasons. It makes it assured that your computer can access the network, do what it needs to do, and function how they need it to at a baseline. This way the IT department can maintain it easily, quickly, and up to standard.

Can they make me do this?

YES THEY CAN! They are having you work on their property, with their property, to make sure it works properly. This way you can actually do your job.

Should I do this?

Oh god yes.

• This lets you pass the buck if needed. Now if something that is supposed to work, doesn't work, it isn't your fault.
• Maintenance becomes a breeze because if your files are backed up to a safe place (any installers you used as well), then if something really bad happens they can restore it to a disk image and have your computer back to you in the matter of a few hours instead of days.
• IF you leave the company for any reason, you don't have to relinquish your personal computer to them for driver scrubbing or making sure you don't take any company software or intellectual property with you.
• IT has no claim to touch your personal computer for any reason.
• For security reasons, you can make sure that your work computer is up to their standards and any potential breach won't be your fault, but their bad policies' fault.

And here's the whammy: It keeps you safe from the company!

In using the company computer your own personal information won't be on the company network, and you can keep your private life away from your work life. This is a big advantage because you can make sure that your own data, is your own data.

As a guy who writes and enforces these types of corporate policies, I can tell you this: it is perfectly normal, and a perfectly reasonable policy.

I do NOT want your equipment on my network, ever. I can't control it, I have no insight into how patched it is, how virus-ridden it is, and I do NOT want you to keep company data on your personal device when you leave.

The sheer number of things that have gone horribly wrong when an employee comes in on the weekend with their personal laptop so they can do personal stuff on it while working on their work laptop makes me shudder. One laptop was a host for 4 different botnets and the owner didn't even know it. My network knew it, though, when the bots started scanning and probing my network.

In the end, you need to ask your IT Security department these questions. They will be happy to describe in detail why their policies are the way they are.

In addition to all the other reasons given:

Software licences. You and other employees need certain programs to do your work. These programs are usually licensed for a limited number of users.

The company want to control this and the easiest way is by controlling the machines. Letting employees install these programs on their own computers would be both a practical and legal mess.