I recently replaced the screens of an iPhone 5S and an iPhone 6S and run into one specific issue that wasn't specified in any of the instructions I used for the process.
[Note 1: This issue doesn't apply to the iPhone 7, 7+, 8, or 8+]
When moving the home button to the replacement screen, make sure that the bracket mounting screws are not tightened much at all. If the screws are too tight the home button will not click. To prevent this, only lightly tighten the screws until they hit resistance and stop.
This will likely feel too loose, but it is correct. Any additional tightening will result in the home button click issue. Click the home button a few times to ensure that it feels right before reassembling the phone.