Better way to switch Brush colours?

If you use the remainder operator (and you assume non-negative input) you know you'll always have a value in the range 0 to 5 inclusive, so you don't need a switch at all - just use an array:

private static readonly Brush[] brushes =
{
    Brushes.HotPink,
    Brushes.Orange,
    Brushes.PeachPuff,
    ...
    Brushes.White
};

// TODO: Potentially rename from using "position" to something else,
// based on what the parameter is really meant to represent.
public Brush GetBrushForPosition(int position) => brushes[position % 6];

If the input might be negative, you can use the slightly more long-winded expression of ((position % 6) + 6) % 6 for the array index, which will still cycle appropriately. (There are other approaches of course, but that's reasonably simple.)

You could reduce the number of case labels using the modulus operator:

public Brush Colour(int i)
{
    Brush snakeColour;
    switch (i % 6)
    {
        case 0:
            snakeColour = Brushes.HotPink;
            break;

        case 1:
            snakeColour = Brushes.Orange;
            break;

        case 2:
            snakeColour = Brushes.PeachPuff;
            break;

//      etc.

        default:
            snakeColour = Brushes.White;
            break;

    }

    return snakeColour;
}

But note: this would most likely remove the need for your default case (assuming you have all cases from 0 thru 5 taken care of). That may be contrary to what you intend, if the actual i value is outside a specific range!

By using % operator, you can make your code much more simpler. Here is what @Adrian Mole suggested...

 public Brush Colour(int i)
        {
            Brush snakeColour;
            i %= 6;
            switch (i)
            {
                case 0:
                    snakeColour = Brushes.HotPink;
                    break;

                case 1:
                    snakeColour = Brushes.Orange;
                    break;

                case 2:
                    snakeColour = Brushes.PeachPuff;
                    break;

                default:
                    snakeColour = Brushes.White;
                    break;

            }
            return snakeColour;
        }