I just upgraded from UBB Classic, and our old board required people to login to see any pages. Is there a way to enable this on Threads?

Set permissions for "guest" to none for each forum. Note that if you ban guests from viewing your forums, you will also be banning search engine spiders from crawling your content.

Good, I want that too.

That's got part of it, now is there a way so they can't see the islands, like shout box, calendar, etc?

That is somewhat limited but you can see your options at Control Panel » Group Management » Site Permissions

Under UBB Classic there was an option under "Display Options"/"General". On the "Primary" tab, expanding "General Options" there was a checkbox option for "Restrict Access to Registered Users". Is there anything like that in threads? Does this variable still exist?

Okay I remember that setting you want a completely closed board from unregistered visitors,Correct? Remember no spiders.
This is more cumbersome the first time in the new version. Not a simple one click .Several steps to take

First Goto:
Control Panel » Group Management
Edit the guest group for Forums and place a 0 in every single item
Then go back to Control Panel » Group Management
Select Guest group for site and again Place a 0 in every single item

Then goto Control Panel » Portal Settings
And unselect "Enable the Main Portal Index page? "

Guests will only see the login page period
Of course that means you can't use the portal page.
Unless someone else chimes in with a way to do it.

I guess I'll just change the code myself. I'll try adding PHP code to hide left and right columns if the person isn't logged in, and redirect all pages to the login/register pages. Maybe I'll go search UBBDev, I can't be the first person who wanted a completely private forum.

simple enough change.. libs/html.inc.php

function: send_column

just add a check for EACH island versus being logged in.. there is already some that are scrutinized this way, so you have a good starting point

In case you misunderstood about the portal page.
The Portal page is a public view item. If you turn it off, You still have the islands appear when logged in and view the forum summary page or forums.