I'm putting labels on a county map. I made a line layer called 'County Borders' with all the neighboring counties. I want to keep those labels on the outside of the county border, while keeping other point and road labels inside the county.
Working with a CO County boundary feature class and as a polygon vs a line, the Labeling contextual tab does have a Boundary (polygon) label placement option. In addition, for the Maplex labeling engine, within the position group, there is a Feature Weight for Interior placement and Boundary placement. I wonder if you convert your line feature class to a polygon feature class and use the label placement options for the polygon, will that get you closer to your desired labeling outcome?
To keep the neighboring county labels outside the focus county (and inside their own county), @Robert_LeClair's suggestion of using Boundary polygon placement is the easiest.
If that's not an option and you must work with the line features, you can use the Constrain offset property of the offset line labels, set to either Left of line or Right of line, depending on which direction the lines are digitized.
Add a high Feature weight to the county lines layer to prevent other labels from overlapping the county lines.