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there isn't a way to get them back if you save/overwrite.Of course there is, in your case. If patient identifiers have a fixed, known length, then you can pad with leading zeros to recover them.
You only have a problem if 012345 and 12345 are distinct patient identifiers.
It is bone-headed in the first place to use numeric-looking identifiers (such as containing digits only) which are really strings, and then allow leading zeros. Identifiers which are really strings should start with a letter (which could be a common prefix). E.g. a patient ID could be a P00123.
This is useful for more than just protecting the 00. Anywhere in the system, including on any printed form, if you see P00123, you have a clue that it's a patient identifier. An input dialog can reject an input that is supposed to be a patient identifier if it is missing the leading P, or else include a fixed P in the UI to remind the user to look for P-something in whatever window or piece of paper they are copying from.
The main point in my comment is that instead of shaking your fist that the behavior of other users in the system, such as those who choose Excel because it's the only data munging thing they know how to use, you can look for ways that your own conventions and procedures are contributing to the issue.
If people are going to "Excel" your data, and then loop it back to you, maybe your system should be "Excel proofed".