Can I change a field type after a form has been published and not lose the data

If you're referring to replace a current field in your form with another one, for example, replace a check box with radio buttons, then this is not possible. 

You will need to remove the field in order to add the new question, but with different fields.

A way that I can see to achieve the same without deleting the current field in the form, would be hiding this field and adding the new question, using the new field, so you will not lose the data previously sent and will get the data from the new field, but as you will hide the old field, it will be sent as empty, but you should not be afraid about that, empty fields are hidden in the submissions page or email notifications. 

To hide the field in the form follow these steps. 

First click on the field, click on the gear icon and then click on the option to hide the field: 

This is the only way that I can see to replace a field without losing the data. 

Hope this helps. 

Regarding to hidden fields on submissions or email notifications, they are hidden by default, but in case of emails, you should not modify the email template, if you do it then empty fields will not be hidden anymore on emails.