I guess it depends on what you are trying to achieve. Do you want to be able to track which user has clicked to increment the number?
Also if you have thousands of users updating the same value you need to make sure that you are handling events in the case where 2 users click at the same time.
I reckon it depends on what you are trying to achieve in the end, if you want to track and report on the user's behaviors as well as not having to deal with multiple simultaneous clicks I would suggest you go for a record per user.
If the size of the database would be an issue some how you would probably have to write some queue-ing logic to ensure that all users' clicks get registered and then you could use the same value.
That being said, I would strongly suggest to use a record per click as it's easier to handle as well as provides roll back capabilities