I think it's just factually inaccurate to say that X President isn't "my" President if your a citizen so it doesn't make much sense to say it. Just like Dick Durbin and Mark Kirk are my senators, Obama is my president.
I do not think, however, that saying, for example, "Barack Obama is not my president," is necessarily an example of putting "ideology over country." The problem is that you're equating an elected official with the country when, in fact, the country is much more than any elected official including the President. Even further, if someone who you believed was truly harming the country is President, then rejecting them as your President may be putting your country first. This is why we have the impeachment process.
Summary: Don't equate the President with the country. They are two separate entities. Moreover, rejecting a President may be better for the country than doing otherwise if that President causes harm to the country.