How to solve 7 blue boxes

Let $x$ denote the number of blue boxes that get filled and $y$ the number of red boxes that get filled. Then there are $5x$ red boxes and $5y$ green boxes. Hence,

The total number of boxes is $$7+5x+5y=7+5\cdot(x+y)$$ and the number of empty boxes is $$\underbrace{7-x}_{\text{empty blue boxes}}+\underbrace{5x-y}_{\text{empty red boxes}}+\underbrace{5y}_{\text{empty green boxes}}=7+4\cdot(x+y)=87.$$

Hence, $x+y=20$ and thus there will be a total of $7+5\cdot 20=107$ boxes. So your guess was correct - congratulations!