I misread the question and wrote a bad answer, so here's a new one...
I assume this is a typical client-server-db setup. You need to obtain the local time zone from the client that's clocking in/out the user; Postgres doesn't know what it is. We'll figure out the user's local date from that and store it. Then we'll have a uniqueness index on <user, local date>.
I thought there'd be fancier ways to do this by storing the timestamptz with a separate time zone col and calculating the date within the uniqueness index, but Postgres doesn't allow us to use date_trunc within an index. So we're going to denormalize just a little and make things a lot easier with this additional date col.
CREATE TABLE clock_in (
user_id bigint NOT NULL,
created_at timestamptz NOT NULL, -- stores microseconds since epoch
local_date date NOT NULL, -- stores the <year, month, day> in whatever timezone the user clocked in from
-- optional for bookkeeping purposes: time_zone text NOT NULL,
UNIQUE(user_id, local_date)
);
Take a look at the official date/time type docs for further explanation of the above. IMO you shouldn't rely on DB constraints to reject bad user input. They're more of a second line of defense meant to ensure a self-consistent database. First your server should query the last clock-in and error out if it was in the same day, and also error if there was no clock-in that day. You'll be able to yield more useful error messages that way. Then you can insert...
INSERT INTO clock_in(user_id, created_at, local_date) (
SELECT 1, now(),
(date_trunc('day', now() AT TIME ZONE 'insert_users_timezone_here'))::date
);
Usage example for a client who has indicated it's in the PST timezone:
me=# CREATE TABLE clock_in ( user_id bigint NOT NULL, created_at timestamptz NOT NULL, local_date date NOT NULL, UNIQUE(user_id, local_date) );
CREATE TABLE
me=# INSERT INTO clock_in(user_id, created_at, local_date) ( SELECT 1, now(), (date_trunc('day', now() AT TIME ZONE 'PST'))::date );
INSERT 0 1
me=# INSERT INTO clock_in(user_id, created_at, local_date) ( SELECT 1, now(), (date_trunc('day', now() AT TIME ZONE 'PST'))::date );
ERROR: duplicate key value violates unique constraint "clock_in_user_id_local_date_key"
DETAIL: Key (user_id, local_date)=(1, 2022-04-13) already exists.
me=# INSERT INTO clock_in(user_id, created_at, local_date) ( SELECT 1, now(), (date_trunc('day', now() AT TIME ZONE 'PST' + interval '10' hour))::date );
INSERT 0 1
me=#
Then you'd do the same for clock-outs.
Using timestamptz instead of timestamp is deliberate. You should almost never use timestamp, for reasons other answers describe well.