Options not working

That sounds like a JavaScript problem. Open the JS console and check if any error messages show up when you click one of those options.

Here’s how to open the console:
https://webmasters.stackexchange.com/questions/8525/how-to-open-the-javascript-console-in-different-browsers

I’ve done that and this is the message that appears:

Uncaught SyntaxError: Unexpected token ILLEGAL
tools.php?page=view-broken-links:1284

which leads to:

$('#blc-delete-filter').click(function(){
var message = '<br />

Should it read:
var message = '<br />'

There shouldn’t be a <br /> tag there at all. Assuming that you’re using the English version of WordPress, that part of the code should look like this:

$('#blc-delete-filter').click(function(){
	var message = 'You are about to delete the current filter.\n\'Cancel\' to stop, \'OK\' to delete';
	return confirm(message);
});

The message depends on WordPress language settings. If there’s a <br /> tag there, then either there’s a PHP error (are there any error messages in the page source code, below the line you posted?), or the localisation is broken.

Hm, it worked not too long ago and I am using the English version. I just went back and also saw this message listed on the console:

Failed to load resource: net::ERR_CACHE_MISS

This wasn’t there before when I looked. I don’t see any PHP erros.

Thanks for your help.

Should I deactivate the plugin, delete the files, and re-install?

Failed to load resource: net::ERR_CACHE_MISS

That’s probably an unrelated issue.

I don’t see any PHP errors.

How about in the PHP error log? If you don’t have it enabled, here’s how to configure error logging.

Should I deactivate the plugin, delete the files, and re-install?

I’d recommend first first deactivating other plugins (temporarily) to check for plugin conflicts. If that doesn’t help, you could try the reinstall.

Deactivating all other plugins didn’t make a difference in the behavior on the Broken Links page. I reinstalled and am waiting now for the links to be scanned.

Alright, I have broken links showing up again and got error logging set up. This is what I’m getting in the error log:

Dec-2014 12:25:41 America/Denver] PHP Notice: Use of undefined constant ENT_HTML401 – assumed ‘ENT_HTML401’ in /data/web/htdocs/burstaff-dev/wp-content/plugins/broken-link-checker/includes/admin/links-page-js.php on line 797

All right, that’s a bug. BLC shouldn’t use the ENT_HTML401 constant because it was added in PHP 5.4 and WordPress only requires PHP 5.2.

I’ve fixed the notice in the development version.

This is fixed after the update, thank you!