Query that ignore the spaces

It depends. If you don't care about good performance then there are lots of things you could do but most of them will be slow. Maybe that's OK for you, but I will leave this answer here in case others reading do want a fast solution.

If you want very fast performance you should index the string without spaces in the database. In PostgreSQL you can create an index on a function. You can use this to create an index on the column with spaces replaced with the empty string. The advantage of this method is that it requires no maintenance apart from creating the index.

In MySQL you cannot do this so the simplest way would be to duplicate the data in the database - once with spaces and once without. Use the column without spaces in your WHERE clause, but the original column in your SELECT column list. This requires more maintenance as the columns must be kept in sync. You can do this with application logic or database triggers.

The proposed solution look very well but is horrible for performance, if it is possible restrict the query with something like:

SELECT * FROM mytable 
    WHERE username like 'John%' and REPLACE(username, ' ', '') = REPLACE("John Bob Jones", ' ', '')

Also you can use REGEXP:

SELECT * FROM mytable 
    WHERE username REGEXP '^John *Bob *Jones'

And remember the performance, operation in the where are in general bad idea.

Take a look to http://dev.mysql.com/doc/refman/5.7/en/pattern-matching.html

TRY THIS:

SELECT * FROM mytable WHERE username =REPLACE("John Bob Jones", ' ', '')

SELECT * FROM mytable 
    WHERE REPLACE(username, ' ', '') = REPLACE("John Bob Jones", ' ', '')