Managed to fix this (for anyone that stumbles across a similar issue in the future!). The issue was that the figure was only displaying three ticks (the final 3) and so when it went to assign tick labels, it only assigned them to the final three. I inserted the final line:
boxplot(Data(:,1:3),'notch', 'on','Colors',"r","Positions",X1)
boxplot(Data(:,4:6),'notch','on','Colors',"b",'Positions',X2)
boxplot(Data(:,7:9),'notch','on','Colors',"g",'Positions',X3)
xticklabels({'1mm A','1mm B','1mm C','1.5mm A','1.5mm B','1.5mm C','3mm A','3mm B','3mm C'})
set(gca,'xtick',1:9); %inserted this line!!
Which gave the plot 9 ticks, allowing all 9 tick labels to be displayed.