Printing rectangular grid
This sounds like you have an adhesion problem if it catches laid down filament, you might want to address that first. E.g. use a PVA based glue or spray to get better adhesion. This will result in not dragging laid down filament. 

To my knowledge, Ultimaker Cura has no option to choose how you print the squares (direction and start point). However, you could use Z-hop so that it will lift your nozzle (or lower your build plate in your case) prior to moving to the next rectangle.

It looks as though you are using an older version of Ultimaker Cura as it only prints one line of each of the small rectangular holes (or are you actually using a single wall/shell), in later versions of Ultimaker Cura this is fixed (e.g. the image below is created with Ultimaker Cura 3.4.1), it will print all walls/shells before commencing to the next small rectangular hole. This way you have more lines deposited which have a possible better adhesion to the increase of laid down material:



Further investigation of your image shows that you are using a very fine grating (&lt; 1 mm?) resulting in very limited amount of walls. In your case the version is not that important, but the latter information is just left as a possibility for people that use an older version of Ultimaker Cura.

Also note that there is an option to put the brim on the inside of your models (option called Brim Only on Outside), when disabled, this would also increase the surface area for better adhesion.



There are also option available to start with the outer or inner wall (option Outer Before Inner Walls), but in this case that would not help you as there is only 1 wall at each side of the rectangular hole.