Examine the area of linoleum that you are cutting the square out of to determine if there are grout lines or a pattern on it, which helps to disguise the cuts. If the linoleum consists of square tiles, you will simply remove the tile.
Place a straight edge at the left side of the damaged areas, aligning it with the pattern you located. If you have linoleum tiles, align the straight edge with the left side. Hold the straight edge down firmly with one hand and place a utility knife at the top of it. Pull the knife blade down past the area you want to remove.
Reposition the straight edge at the top of the original cut, perpendicular to the end. Cut along the straight edge with the utility knife to create the top of the square. Repeat the process on the right side and the bottom of the area, until you have cut a complete square.
Plug in a hair dryer and turn it to the highest heat setting. Aim the air directly at the square and hold it for at least 60 seconds to soften the flooring adhesive.
Insert a putty knife under one corner of the square and push it firmly under the square to pry it up off the subfloor. Continue prying until the entire square is removed.
Scrape the subfloor firmly with the putty knife to detach the excess glue and then proceed with installing a new piece of linoleum.