Sudoku solver recursive backtrack in Java
Object Orientated Programming

you don't use object orientated programming! that would clean up the code in a large scale

public class Board {
    //private int[][] board; primitive obsession - use objects instead
    private Field[][] fields;
    private Column[] columns;
    private Row[] rows;
    private Block[] blocks;
    public final static int SIZE = 9;  

    ...      
}


you could provide the proper methods for the field - so you could easily represent the field - instead of trying to interpret an int value.

you could provide methods that are specific for these objects - you could even define an interface for common code, like isFieldValid(Field candidate) for Row, Column and Block 

Naming

consider these names and you might find better

Random rand = new Random();
int random, randomrow, randomcolumn;


maybe it would be 

Field candidate;


again here, in Java we don't use the hungarian notation and name a variable by it's type:

//int[][] boardarr = board.getBoard();
int[][] board = board.getBoard(); //would be better
Field[][] fields = board.getFields(); //even more better


Data Structure

if you would use Objects for your Sudoku-Solver you could re-use values (here an example snippet for Block)

class Block {
    final private Set&lt;Field&gt; fields = new HashSet&lt;&gt;();

    boolean isFieldValid(Field field) {
        //Set&lt;Integer&gt; set = new HashSet&lt;&gt;(); we already have such set 
        //and don't need to create a new one whenever we check
        return !fields.contains(field);
    }
}


Magic numbers

you already provide a SIZE = 9 constant - so why don't you use a proper one for the block size?

int blockrow = (row / 3) * 3;
int blockcolumn = (column / 3) * 3;


Testing

i assume you provide enough tests but just don't put these in the question

Summary

very nice piece of code - i'm glad you have posted it here! The algorithm seems straight forward, i like it! The OO-thing is the only flaw here. I would appreciate if you would provide another question with applied OOP!