Object Literal Pattern in Typescript
I'm not going to suggest any major Typescript features you aren't using, as it looks like you don't need any here. Additionally, I'm not going to claim any of my following questions / suggestions are "best practices" as "best practices" seem to change annually at a minimum in the web development world. 



Why are you defining an interface for a single object? I don't see any reason to burden yourself with maintaining multiple definitions. You can include the method definitions within the module itself to save yourself from scrolling back up to the interface to check what the definition should be. 

This is more of a stylistic preference, but I see no need to preface your methods with square. Since you are accessing the method on the squareWidget object it seems redundant to include square a second time. 

When squareWidget is created, the list of squares is immediately created. This is a problem as if you include your code before the HTML has finished loading, the squares added before calling squareWidget.init(), but after the code is parsed will be missed. 

In your squareSizer function, you use a dataset value which is not set by the init function. If the value is missing, the method won't do anything. When creating a square, should I have to remember to specify if it is enlarged or not every time? Also, is the dataset used in any code not included in your post? If not, you could replace your checks with the built in classList.toggle()

In squareColorChanger, instead of looping through the classes on the element, it might be better to check for and remove the color classes specified by the module. You can even take this a step further and just use el.classList.remove(...squareWidget.squareColors)

In doSquareClick I expected to see code that handled a click. Instead, the method adds an event listener to the element. Perhaps addListener would be more descriptive? In addition to this, your method asks for more arguments than it uses. You can safely remove the index and array arguments with no effect to your code. 

In init the Array.from is unnecessary in newer browsers. forEach is included on NodeLists. Despite this, I would leave it as is until IE dies out completely. One concern I have with the current function is that if I call it twice, two event listeners will be added on all elements. If I add squares to the page after it has loaded due to user interaction I need to call the init method again to activate them which then breaks the existing squares. You could solve this problem several ways. My personal choice would likely be defining a single event listener method which is added on all squares. With this change, calling addEventListener again will have no effect.