I would like, that the user is the only one who can change the checkbox, but only set it to false. Where is my mistake? He can only set to TRUE.

the formula will be as below assuming you want this user to set only the checkbox to False

AND( 
Checkbox__c = TRUE,
ISCHANGED(Checkbox__c), 
$User.UserName = "Insert Excluded UserName here"
)

but in the last line you have mentioned He can only set to TRUE., could you please clarify what the actual requirement is.

If the requirement is user can only set the checkbox to True but not unchecked or False then the formula will be

AND( 
Checkbox__c = FALSE, 
ISCHANGED(Checkbox__c), 
$User.UserName = "Insert Excluded UserName here" 
)

I would not use UserRole Name as suggested by Naveen and Girish as you can have more than one user in the same role