Hi Thomas,
Thanks for using Syncfusion products.
You can achieve your requirement by using CustomCss propery. The three parts of the Upload box which refers the below class
.UploadSubmitButton - refers to the Upload button.
.UploadButton - refers to the Browse button.
.UploadTextCont - refers to the text field.
Also I have created the simple sample with this and attached in the below link for your reference,
UploadBox224006087.zipPlease try this and let us know if you have any queries.
Regards,
Vignesh