What is the recommended collation for a postgresql citext column?

That citext doc already tells you it's somewhat superseded by case-insensitive collations:

Consider using nondeterministic collations (see Section 23.2.2.4) instead of this module. They can be used for case-insensitive comparisons, accent-insensitive comparisons, and other combinations, and they handle more Unicode special cases correctly.

You're better off with a regular text type and a custom collation, or "C" with an expression index using lower(). You can find a few benchmarks here:

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If you upgrade to version 18 (release candidate 1 is out), you get nondeterministic collation support for LIKE which handles your prefix search.
In PostgreSQL 16, use collate "C" with a text_pattern_ops expression index:
_{demo at db<>fiddle}

create unique index on test_lower_collate_c 
  (lower(a) collate "C" text_pattern_ops);

explain analyse verbose
select count(*) from test_lower_collate_c where lower(a) like 'eb%5%';

Will leaving the collation on these columns as C cause issues, since C is a case sensitive collation?

Values get folded to lowercase when ingested into citext type so the case differences are lost - that's not a problem.

It might be a problem if you're dealing with accents and other non-ASCII texts because collate "C" places them in a different range. According to it where a ~>=~ 'ea' and a ~<~ 'eb' won't find values starting with 'eá' because accent variants go somewhere way behind the whole alphabet instead of following their base letter.

Another thing of note is that I don't see the optimiser adding the range scan to pattern-based search on its own. Given a query like this:
_{demo at db<>fiddle}

select from test_lower_collate_c where lower(a) like 'eb%5%';

text_pattern_ops gets you an additional condition to speed up the search based on prefix

Index Cond: ((lower(test_lower_collate_c.a) ~>=~ 'eb'::text) AND (lower(test_lower_collate_c.a) ~<~ 'ec'::text))

Meanwhile, with citext_pattern_ops I needed to add them on my own:

select from test_citext_collate_c where a like 'eb%5%' and a ~>=~ 'eb' and a ~<~ 'ec';;

And the timing was still worse than for the expression-based index.

What is the recommended collation for a citext column?

If your values/patterns are simple ASCII, COLLATE "C" can handle them fast. Otherwise, it just won't work right.