Why can't I leave blank the labels that are shown on a checkbox field?

 If you have multiple options and try and make the option name blank, the checkbox disappears and won't let you edit it again.  It also appears it won't allow you to have a blank checkbox, which is dumb.

I presume that you're referring to the label that is shown on each checkbox, is that correct?
If so, the reason why you cannot leave them blank is because the label is used to let you know which options were checked.

If you don't want them to be visible in your form, you can hide them using custom CSS code. For this form: http://www.jotformpro.com/form/51365821556963 you can use the following code:

#cid_3 > div > span > label {
   display: none;
}

This would be the result:

Hope this helps.

Let us know if you need further assistance.