First of all, the existing code you shared has a few problems.
Missing Closing Brace for Rectangle
The scope of the Rectangle class is not closed (you're missing a closing curly brace "}") after the last function.
The corner1 and corner2 variables in the constructor are undefined. Remove them and now your Rectangle class (placed in Rectangle.java) should compile.
Constructor For a Square
Modify the Rectangle class to permit construction of rectangles using a single input argument (in which case both height and width are to the input value).
describes a constructor for a square (where both width and height are the same).
Look at the existing constructor (public Rectangle(int x, int y)), make a copy of it and remove one of the parameters. Then in the body of your freshly created constructor set both the width and height to be the only parameter of the new constructor.
Printing The Perimeter
I suggest doing this first as it is easier than the diagonal.
Make a copy of one of the existing functions other than the constructor, e.g.: public void printDims(). Change the name to the desired printPerimeter. Then in the body of the function, change the text to be printed to say it is the perimeter (instead of the width and height).
Calculate the perimeter (2 * (width + height)) of the rectangle and print that instead of printing the height and width.
Printing The Length of The Diagonal
Make a copy of one of the existing functions other than the constructor, e.g.: public void printDims(). Change the name to the desired printDiag. Then in the body of the function, change the text to be printed to say it is the diameter (instead of the width and height).
You will need to create a new variable to store the diameter. I suggest using a type that supports fractions, so use a floating-point number (double).
You will need to use the Pythagoras Theorem to calculate the diagonal. If you need help with this, check this out. For calculating the square of a number, just multiply it by itself, i.e., x2 can be written like this: x * x. To calculate the square root of x, you can use the Math.sqrt(x) expression.
It's Your Turn
I hope these instructions can get you started.
Get your hands dirty, and write some code! Give it your best shot. If you get stuck, post what you have so far and people will be happy to help you move forward.